Jae is constructing an open box from a piece of cardboard that is 9 in. wide and 12 in. long. He cuts squares of equal size from each comer of
the cardboard, as shown in the picture, and then folds up the sides of the box. Write and simplify a polynomial function V for the volume of the box in terms of x
Answers
Answer:
The polynomial function for the volume of the box ([tex]V[/tex]) in terms of [tex]x[/tex] is
[tex]V(x)= 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex].
Step-by-step explanation:
We present a representation of the specifications of the open box in a image attached below. The volume of the open box ([tex]V[/tex]), measured in cubic inches, is represented by this expression:
[tex]V = w\cdot h \cdot l[/tex]
Where:
[tex]w[/tex] - Width, measured in inches.
[tex]h[/tex] - Height, measured in inches.
[tex]l[/tex] - Length, measured in inches.
Polynomial functions in standard form are represented by the following form:
[tex]y = \Sigma_{i=0}^{n} c_{i}\cdot x^{i}[/tex]
Where:
[tex]n[/tex] - Order of the polynomial, dimensionless.
[tex]c_{i}[/tex] - i-th Coefficient, dimensionless.
[tex]x[/tex] - Indepedent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
If we get from figure that [tex]w = 12 - 2\cdot x[/tex], [tex]h = x[/tex] and [tex]l = 9 - 2\cdot x[/tex], then:
[tex]V = (12-2\cdot x) \cdot x \cdot (9-2\cdot x)[/tex]
[tex]V = (12\cdot x -2\cdot x^{2})\cdot (9-2\cdot x)[/tex]
[tex]V = 108\cdot x-18\cdot x^{2}-24\cdot x^{2}+4\cdot x^{3}[/tex]
[tex]V = 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex]
The polynomial function for the volume of the box ([tex]V[/tex]) in terms of [tex]x[/tex] is
[tex]V(x)= 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex].
The volume of a box is the amount of space the box can occupy.
The polynomial function is: [tex]\mathbf{V(x) = 4x^3 -42x^2 + 108x }[/tex]
The dimension of the cardboard is given as:
[tex]\mathbf{Length = 12}[/tex]
[tex]\mathbf{Width = 9}[/tex]
Assume the size removed from the box is x
The dimension of the box becomes
[tex]\mathbf{Length = 12 -2x}[/tex]
[tex]\mathbf{Width = 9 -2x}[/tex]
[tex]\mathbf{Height = x}[/tex]
So, the volume (V) of the box is:
[tex]\mathbf{V = Length \times Width \times Height}[/tex]
Substitute known values
[tex]\mathbf{V = (12 -2x) \times (9 - 2x) \times x}[/tex]
Open brackets
[tex]\mathbf{V = (12 -2x) \times (9x - 2x^2)}[/tex]
Open brackets
[tex]\mathbf{V = 108x -24x^2 - 18x^2 + 4x^3}[/tex]
[tex]\mathbf{V = 108x -42x^2 + 4x^3}[/tex]
Rewrite as:
[tex]\mathbf{V = 4x^3 -42x^2 + 108x }[/tex]
Express as a function
[tex]\mathbf{V(x) = 4x^3 -42x^2 + 108x }[/tex]
Hence, the polynomial function is:
[tex]\mathbf{V(x) = 4x^3 -42x^2 + 108x }[/tex]
Read more about volumes and functions at:
https://brainly.com/question/1758167
Related Questions
i feel so stupid right now it makes me mad
how do you describe a correlation
Answers
Answer:
Correlation is a statistical technique that is used to measure and describe a relationship between two variables. Usually the two variables are simply observed, not manipulated. The correlation requires two scores from the same individuals. These scores are normally identified as X and Y.
Step-by-step explanation:
What is the range of the function f(x) = 4x + 3 when the domain is {−3 , 2 , 4}
Answers
Hey there! I'm happy to help!
The range of a function is all of the outputs. Let's plug in our inputs and find these outputs.
4(-3)+3
-12+3
-9
4(2)+3
8+3
11
4(4)+3
12+3
15
We order these from least to greatest, giving us a range of {-9,11,15}.
Have a wonderful day! :D
Solve for x. Make sure to show all your work.
Answers
Answer:
x=18
Step-by-step explanation:
(6x+3)+(3x+15)=180
(6x+3x)+(3+15)=180
9x+18=180
-18 -18
9x=162
162/9
x=18
5) Answer the following questions :
The total number of boys and girls in a school is 320. the
number of boys in this school is 180 find :
a) The ratio between the number of boys and that of girls.
b) The ratio between the number of boys & the total number of
pupils.
c)The ratio between the number of girls & the total number of
pupils.
Answers
Answer:
a)9:7 b) 9:16 c) 7:16
Step-by-step explanation:
the no of girls (320-180)=140
a) ratio of no of boys and girls=180/140=9/7=9:7
b)ratio of boys and total pupil=180/320=9/16=9:16
c) ratio of the girls and pupils=140/320=7/16=7:16
Answer:
a) 9:7 (9 being the boys,7 being the girls)
b)9:16 (9 being boys, 16 being the total pupils)
c)7:16 (7 being girls, 16 being the total pupils)
Step-by-step explanation:
a) No. of girls= 320-180
=140
Boys: Girls
180 : 140 [ ÷ 20]
9 : 7
b) Boys: Pupils
180 : 320 [ ÷ 20]
9 : 16
c). Girls: Pupils
140 : 320 [ ÷ 20]
7 : 16
Point G is the incenter of the triangle. Point G is the incenter of triangle E D F. Lines are drawn from each point of the triangle to point G. Lines are drawn from point G to the sides of the triangle. Angle D F G is 16 degrees. Angle G F E is (2 x) degrees. What is the value of x? 4 8 24 32
Answers
Answer:
8
Step-by-step explanation:
8
Answer:
B
Step-by-step explanation:
2x=16
16/2=8
Answer is 8
what does 5/8 x 30 1/2 = in mixed number form
Answers
Answer:
19 1/16
Step-by-step explanation:
Answer:
305/16
Step-by-step explanation:
Simplify the following:
(5 (30 + 1/2))/8
Hint: | Express (5 (30 + 1/2))/8 as a single fraction.
(5 (30 + 1/2))/8 = (5 (30 + 1/2))/8:
(5 (30 + 1/2))/8
Hint: | Put the fractions in 30 + 1/2 over a common denominator.
Put 30 + 1/2 over the common denominator 2. 30 + 1/2 = (2×30)/2 + 1/2:
(5 (2×30)/2 + 1/2)/8
Hint: | Multiply 2 and 30 together.
2×30 = 60:
(5 (60/2 + 1/2))/8
Hint: | Add the fractions over a common denominator to a single fraction.
60/2 + 1/2 = (60 + 1)/2:
(5 (60 + 1)/2)/8
Hint: | Evaluate 60 + 1.
60 + 1 = 61:
(5×61/2)/8
Hint: | Express 5×61/2 as a single fraction.
5×61/2 = (5×61)/2:
((5×61)/2)/8
Hint: | Express ((5×61)/2)/8 as a single fraction.
((5×61)/2)/8 = (5×61)/(2×8):
(5×61)/(2×8)
Hint: | Multiply 2 and 8 together.
2×8 = 16:
(5×61)/16
Hint: | Multiply 5 and 61 together.
5×61 = 305:
Answer: 305/16
Which of the following is equivalent to 3⁴? 1. 12 2. 81 3. 7 4. 64
Answers
Answer:
81
Step-by-step explanation:
what is 1 1/3 divided by 3/5
Answers
rounded answer: 2.23
Answer:
6.11
Step-by-step explanation:
First divide 11by 3 and 3by 5.And divide the answer of 11by 3 by 3by5
Which equation below would be perpendicular to the line x = 5 and goes through the point (2, 6)?
Answers
Answer:
this also I do not know plz tell answer is will also practice
If the salesperson made $2,500,000 in total sales, is their yearly salary $205,000? Yes or No.
Answers
Find point A ( x, y ) if point B is ( 6 , 10 ) and the midpoint between A and B is ( -4,5)
Answers
Answer:
(14,0)
Step-by-step explanation:
slope 10/5 or 2
can someone pls help! is this correct?
Answers
Answer:
It's wrong
Step-by-step explanation:
Answer:yes it is correct
Step-by-step explanation:
if 28 men can complete a work in 16 days wiil 8 men take to complete that work is it a.14 b.48 c.56 d.128
Answers
Answer:
56
Step-by-step explanation:
-4x+1=2x-17 What is the solution for x ?
Answers
Answer:
[tex] \boxed{ \bold{ \huge{ \boxed{ \sf{x = 3}}}}}[/tex]
Step-by-step explanation:
[tex] \sf{ - 4x + 1 = 2x - 17}[/tex]
Move 2x to left hand side and change it's sign
Similarly, move 1 to right hand side and change it's sign
[tex] \dashrightarrow{ \sf{ - 4x - 2x = - 17 - 1}}[/tex]
Collect like terms
[tex] \dashrightarrow{ \sf{ - 6x = - 17 - 1}}[/tex]
The negative integers are always added but posses the negative ( - ) sign
[tex] \dashrightarrow{ \sf{ - 6x = - 18}}[/tex]
Divide both sides by -6
[tex] \dashrightarrow{ \sf{ \frac{ - 6x}{ - 6} = \frac{ - 18}{ - 6}}} [/tex]
Calculate
[tex] \dashrightarrow{ \sf{x = 3}}[/tex]
Hope I helped!
Best regards! :D
which set of angles shares rayAF as a common side?
Answers
Can someone help me with this please
Answers
===========================================
Explanation:
For now, ignore the variables. Let's just focus on the coefficients 12 and 16.
The GCF of 12 and 16 is 4 because this is the largest factor they have in common.
12 = 4*3
16 = 4*4
You could create a factor tree to get the prime factorization of each number
12 = 2*2*3
16 = 2*2*2*2
Each factor tree has two copies of '2' leading to the GCF 2*2 = 4.
--------
Now let's go back to the variables. The term 12n and 16w^3 do not have any variables in common. So the final answer won't have any variables in it. If both terms had say a 'w' in them, then w would somehow be involved in the GCF. But that's not the case here. So we just stick to 4 as our answer.
--------
Side note: If you had 12n+16w^3 and you wanted to factor, then you would factor out the GCF 4 to get 12n+16w^3 = 4(3n+4w^3). Use the distribution rule to confirm this.
(52) + (3+4i) is
O 3 +9i
O 12
O 3ti
O 92 + 3
Answers
Answer:
O 3 +9i
Step-by-step explanation:
What is the solution set for 5(1−p)>−3p−5?
Answers
3p-1=5(p-1)-2(7-2p)
Simplifying
3p + -1 = 5(p + -1) + -2(7 + -2p)
Reorder the terms:
-1 + 3p = 5(p + -1) + -2(7 + -2p)
Reorder the terms:
-1 + 3p = 5(-1 + p) + -2(7 + -2p)
-1 + 3p = (-1 * 5 + p * 5) + -2(7 + -2p)
-1 + 3p = (-5 + 5p) + -2(7 + -2p)
-1 + 3p = -5 + 5p + (7 * -2 + -2p * -2)
-1 + 3p = -5 + 5p + (-14 + 4p)
Reorder the terms:
-1 + 3p = -5 + -14 + 5p + 4p
Combine like terms: -5 + -14 = -19
-1 + 3p = -19 + 5p + 4p
Combine like terms: 5p + 4p = 9p
-1 + 3p = -19 + 9p
Solving
-1 + 3p = -19 + 9p
Solving for variable 'p'.
Move all terms containing p to the left, all other terms to the right.
Add '-9p' to each side of the equation.
-1 + 3p + -9p = -19 + 9p + -9p
Combine like terms: 3p + -9p = -6p
-1 + -6p = -19 + 9p + -9p
Combine like terms: 9p + -9p = 0
-1 + -6p = -19 + 0
-1 + -6p = -19
Add '1' to each side of the equation.
-1 + 1 + -6p = -19 + 1
Combine like terms: -1 + 1 = 0
0 + -6p = -19 + 1
-6p = -19 + 1
Combine like terms: -19 + 1 = -18
-6p = -18
Divide each side by '-6'.
p = 3
Simplifying
p = 3
which phrase should be inserted on the line to correctly compare the two factions 4/5 + 16/20 a is greater than B cannot be compared C is less than and D is equal to
Answers
First, let's reduce our fractions to their simplest forms.
4/5 is already in its simplest form
16/20 = 4/5.
Now that both of por fractions are in their simplest forms, let's compare them.
4/5 is equal to 4/5
D is the correct answer.
Answer: C. is less than
Step-by-step explanation: the person above me is wrong
Suppose you are deciding whether to take a train or a car to travel 3,000 miles across a country. The train travels at 95 kilometers an hour, while a car will travel 65 miles per hour on average. Which mode of transportation will take less time? Note that 1 mi=1.6 km.
Answers
Answer:
The car
Step-by-step explanation:
Train:1mi = 1.6 km
95km = 95/1.6 = 59.375 km
95km/h = 59.375mi//h
Car65mi/h
Comparative:
65 > 59.375
then:
the car is more faster
Answer:
The answer is A : the car will take less time because it travels at a faster speed. the train will travel at 95/1.6=59.03 while the car travels at 65
Step-by-step explanation:
Please Help! 7+b=b+7
Answers
Answer:
Yes, 7+b=b+7
Step-by-step explanation:
It is correct because the both are the same equation; however, the numbers and variables are just switched around. Therefore, that doesn't change the value of the equation.
Hope this helps!
Answer:all real numbers are solutions
Step-by-step explanation:
Maria babysat for 6 hours on Tuesday, 3 hours on Thursday, and 4 hours on Saturday.
If Maria is paid $6.00 per hour, how much money did she make on these three days?
Answers
Answer:
First, convert 150% to the equivalent decimal 1.50 by dropping the percent sign and moving the decimal two places to the left. Then, multiply 9 by 1.50 to get 13.50. Maria now makes $13.50 per hour.
Step-by-step explanation:
I took the test and this is the sample answer
The required earnings for Maria for the three days is $78.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
here,
Maria babysat for 6 hours on Tuesday, 3 hours on Thursday, and 4 hours on Saturday, and paid $6.00 per hour.
Total earining = 6 × 6 + 3×6 + 4× 6
Total earning = 36 + 18 + 24
Total earning = $78
Thus, the required earnings of Maria for the three days is $78.
Learn more about simplification here:
https://brainly.com/question/12501526
#SPJ2
1936
1937
1944
1945
1946
19
23
30
33
?
Answers
Answer:
44?
Step-by-step explanation:
Maria wants to purchase a shirt for $30. She also wants to purchase a purse for $60. She has to pay 7.5% sales tax on her purchase. What will be the total amount Maria pays including tax.
A. $96.75
B. $6.75
C. $83.25
D. $90.00
Answers
Multiply 90 x 7.5% to get 6.75
Add to get answer
Brainliest?
Write the fraction 12/28 in the simplest form
Answers
Answer: 3/7
Step-by-step explanation:
Answer:
12/28 is 3 /7 is your answer
The graph of the function B is shown below. Find B(2).
1
-1
2
Answers
Answer: 2
Step-by-step explanation:
Answer:
The answer is 2
Step-by-step explanation:
Don't forget to thank and vote
What is the supplement of a 110 degree angle
Answers
Answer: 70 degrees
Step-by-step explanation: The supplement of an angle is the angle that has to be added to make 180°. So, the supplement of 110° is 70° because 110+70= 180.
. Una distribución de probabilidades permite organizar todos los resultados posibles de
una variable aleatoria junto con sus correspondientes probabilidades de ocurrencia.
El valor esperado E(x) de una variable aleatoria discreta se calcula multiplicando
cada valor que esta puede asumir por su probabilidad de ocurrencia y luego
sumando los productos.
La siguiente tabla muestra la distribución de probabilidades correspondiente al
tiempo (en minutos) que demora una computadora en recuperarse luego de una
falla. ¿Cuál es el tiempo más probable de espera para volver a utilizar la
computadora?
Tiempo (x) 2.0 3.0 4.0 5.0
Probabilidad P(x) 0.10 0.30 0.10 0.50
Answers
Answer:
i don't understand Spanish ._.
Step-by-step explanation:
man I wish I understand you ._.
Help please!!!!!!!!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
4x=2(x+6)
4x=2x+6
2x=6
x=3
Together Fred and Wilma have saved $345 Wilma has have twice as much money as Fred how much money do they each have define your variables and set up a system of equations with two equations
Answers
Answer:
Fred has saved $115 while Wilma had saved $230
Step-by-step explanation:
Let the amount saved by Fred be F while the amount saved by Wilma be W
Together, they have saved $345
Mathematically;
F + W = 345 •••••••(i)
We are also told that Wilma has have twice as much money as Fred;
Hence; W = 2F ••••••••(ii)
To get the amount of money they both have, we need to solve the equations simultaneously
Substitute ii into i
F + 2F = 345
3F = 345
F = 345/3
F = 115
But W = 2F
W = 2 * 115
W = 230